Solutions of the Dirac-Fock Equations without Projector

Abstract.In this paper we prove the existence of infinitely many solutions of the Dirac-Fock equations with N electrons turning around a nucleus of atomic charge Z, satisfying N < Z + 1 and 
$\alpha\max(Z, N)$<
${2\over {2\over \Pi}+{\Pi\over 2}}$, where 
$\alpha \approx {1\over 137}$ is the fundamental constant of the electromagnetic interaction. This work is an improvement of an article of Esteban-Séré, where the same result was proved under more restrictive assumptions on N.


Introduction
The main topic of this talk is the Dirac-Fock (DF) model. Many quantum models have been studied from a mathematical point of view during this last fifty years, for example the Hartree (H) model, and its more realistic variant, called Hartree-Fock (HF). These studies led to many results, for example the existence of a ground state [7], of infinitely many stationary states [8] for the HF model. This latter is moreover very useful in computational chemistry, where it yields results on the energy levels of atoms, that agree with experimental data, as long as the atomic charge is not too large. The HF model is nevertheless based on a nonrelativistic equation, involving the stationary Schrodinger operator -A. In 1935, Swirles introduced the Dirac-Fock energy functional [10]: this functional approximates the relativistic energy of a system of N electrons moving around a static nucleus of large positive charge Z. There remains some strong approximations: the nucleus is supposed to be fixed (Born-Oppenheimer approximation), the only interactions considered are electrostatic (no magnetic -or spin -interaction), and the wave function of N electrons is represented by a Slater determinant. However, the numerical results obtained with the DF model are in very good agreement with experimental data (see [6], [5], [3]).
Here a ^ -stands for the fundamental constant of the electromagnetic interaction. Their method is variational and, after changes in the functional (smoothing the potential and penalizing the constraint), they reduce it thanks to a concavity property: the major advantage is that the new functional is no more strongly indefinite. Once this reduction is made, they build min-max sequences by a linking argument. Unfortunately, the concavity argument does not work for a(3N -1) > aZc.
Our aim is to improve this result, so that it becomes true for every N and Z satisfying N < Z + 1, amax(Z, N) < aZc.
First note that the result presented here as [4] includes also the case of molecules, involving a distribution of positive charge Z/^, with a probability measure /^, and Z the total positive charge. In the following, we will only consider for convenience a point-like nucleus (the molecular case easily follows).
To set out our result, we first recall the definitions. We will use the system of units such that h = c = 1, and the mass me of the electron is also normalized to 1. The state of a free electron is represented by a wave function ^/(t,x), witĥ (t,.) € ^(^.C 4 ) for any A, which satisfies the free Dirac equation: We denote by ((/>, ip}^2 ^e usual Hermitian product in L 2 (R 3 ,C 4 ). The Dirac equation contains the 4x4 complex matrices ai, as, 03 and /?, which satisfy some algebraic conditions ensuring that Ho is a symmetric operator satisfying (1.2) H^=-^+l.
The DF energy functional, denoted by £, is defined on £^, with where a is positive, with physical value a w ^ p is a scalar and R is a 4 x 4 complex matrix, given by N N (1.9) p(:r) = ^ (^(.r), ^(^)) , R{x, y) = ^ ^(^) ® ^*(y). 1 ^=i Physically, p is the electronic density, R is the exchange matrix.
We look for critical points of £ under the normalization constraint The manifold E and the DF functional are invariant under the action of the group U{N) of N x N self-adjoint complex matrices with determinant of modulus 1. Thanks to this invariance, we may diagonalize the Lagrange multiplier matrix and write the Euler-Lagrange equations in the following form, called Dirac-Fock equations H^k =WA:, where hysically, H^ represents the Hamiltonian of an electron in the mean field due to the nuclei and the electrons. The eigenvalues ^i,...,^ are the energies of each electron in this mean field. With the unit convention, the physically interesting states correspond to 0 < ^ < 1: a positive energy less than the rest mass of the electron.
Our result is an extension of [4, Theorem 1.2]. We only give here a sketch of its proof, which may be found extensively in [9]:

Sketch of the proof
In order to prove this result, we are faced with many difficulties: the corresponding functional is strongly indefinite, there is no compactness property available for this functional, the coulombic potential is hard to deal with, and we need a control of the Lagrange multipliers. To avoid successively these obstacles, we use here many approximation tools so that the final variational problem is simpler to handle with. The strategy can be described like this: (a) In order to get rid of the roughness of the coulombic potential V, w-e replace it with an approximate one, actually a convolution of V with a normalized gaussian depending on a parameter v. Moreover, we define a problem with penalization of the constraint. The choice of the penalization gives directly a lower bound for the energy of the electrons. This step is exactly the same as in [4].
(b) To get some compactness properties, we change the problem into a periodic one: we consider a functional space of periodic functions. We have to change one more time the shape of the potential. The new functional satisfies the Palais-Smale compactness property.
(c) The last problem is that the functional is still strongly indefinite. To deal with this, we operate a Lyapunov-Schmidt reduction to a finite dimensional problem. This reduction is made possible because of the compactness properties of the functional, once steps (a) and (b) are fulfilled. Such a tool was used in a famous paper by Conley and Zehnder [2] on Arnold's conjecture. (e) We pass to the limit as the length of the period goes to infinity. There remains to take the limit of the solutions as the penalization parameter p and the smoothness constant v go respectively to infinity and 0: we will use again the same arguments as in [4].
The first transformation of Problem 2.1 into a more practical one was introduced in [4] (to which we refer for more details): as the coulombic potential V is not a compact perturbation of Ho^ we replace it with a regularized potential, for v > 0: This replacement is made for all terms involving V, i.e. attractive potential and electronic repulsion and exchange terms. We denote by £y the corresponding functional, and the associated one-particle Hamiltonian is denoted by H^.
As announced in (a), we also replace the constraint "<& G E" by a penalization term Ti-p depending on an integer p. We put We actually need a control on the critical points for Ty^, but also on some of its Palais-Smale sequences. Fortunately, we may use a compactness result [4, Lemma 2.1], which indicates the behaviour some Palais-Smale sequences, and ensures that the solutions of Problem 2.2 we will find actually converge to solutions of Problem 2.1. We now have to define a periodic potential corresponding to the coulombic potential. Let Gy^ be the periodic potential defined by its Fourier serieŝ

XII-6
Remark that the coefficients are, for p / 0, gAp} n~t ĉ^ = -W = w • This periodic potential is no more positive, but satisfies Gy^ > -cjCo for a fixed Co > 0. This causes a little change in the computations, relatively to the non-periodic case, but the essential properties are preserved, as u) is close enough to 0.
Thanks to the Fourier series computation, we are able to translate in the periodic case the results of Lemma 1.1, in order to get similar estimates on periodic function spaces.

G^(x -y) [p(x)p(y) -tr(R(x^ y)R(y^ x))} dxdy
This functional is defined in the space The new functional satisfies the Palais-Smale compactness property. Note that, as in the non-periodic case, this functional is invariant by the action ofU(N). We may then define The resolution of this variational framework will give solutions of Problem 2.2 since we get a convergence lemma, which achieves the point (b) of the program.

Lyapunov-Schmidt reduction
So we are led to find critical points for the functional P^ corresponding to the periodic problem. Unfortunately, this functional is still strongly indefinite. One way to deal with this problem is to reduce the functional space to a finite dimensional space, thanks to a Lyapunov-Schmidt method.

XII-7
This reduction is based on the fact that we may split any wave function $ into two components: a "high frequency" component ^u and a "low frequency" one <t>^, where frequency is meant to be the (norm of the) eigenvalue for the free Dirac operator Ho. Thanks to a contraction property of Ho 1^ we solve, using a fixed point theorem, the part of the DF-equations related to the high frequency part: this solution depends of course of the low frequency component.
Let A > 1. We consider the projection A[/ (resp. A^)) on the eigenspaces for Ho corresponding to eigenvalues A with |A| > A (resp. |A| <, A). We get of course HoAu = ^uHo, and HoAo = A^Ho. We put ^u = A^, and ^o = AD<&.
Using the projectors Au and AD on the periodic Euler-Lagrange equation, we get the following system: We observe that, given $ = {(pk) € AoE^ for every fc, a solution to the first equation in 2.13 may be written as S = ((ik)i<k<N^ with ^ € AjjE^ for all fc, and S is a fixed point of the operator T^ mapping (A^E^)^ into itself, given by = % (a, 1 \Au (M^^ + N^^ + ^(i> +=))]) \ L \ °'Pk / J / i<k<N The next lemma shows that we may apply the Banach fixed point theorem to find an unique S associated to $. The following estimate will depend on ^, and we check that the considered <J> + S is not too close to the boundary {<& | Grainy <&=]!}. Lemma 2.4. Let IJL C (0,1). Recall that we have fixed v G (0,1) and p > 3. There exists A > 0 such that, given any A > A and $ C (ApE'^)^ satisfying 0 < Gram^2 $ < (1 -^)fl, (where the projections AD and Au are defined with X), the application X^ defined above, restricted to a ball of radius ^ around 0, is contracting.
Moreover, taking A large enough, the ball {Au{E^)) N D B^2 ).v(0,^) is globally invariant relatively to the action ofl^.
Then the application of the Banach fixed point theorem gives a fixed point for Z^ in the ball {Au(E^)) N n B^JA^O, ^). Thus we may define the map h-^ from (AoE^) N to (A[/£'^) N C\B^2 )-v(0, ^), associating to every <1> the unique fixed point ofZi, in the ball. We put EE = h~^}.
Thanks to this property, we can reduce our problem to a finite dimensional one: if we put, for$ € {AoE(

2.14)
Q^)=p,^+h^)) , then Problem 2.3 is solved when the following problem is solved:

End of the proof
Once this reduction is done, we may use variational tools on the function Q^ acting on the finite dimensional space (AoE^. Briefly speaking, we build a min-max using a linking property. In particular, we have to avoid critical points corresponding to solutions with A/ 7 electrons, for N 1 < N, i.e. critical points $ with a null determinant Gram matrix: this is done by constructing a special pseudo-gradient vector field. We refer to [9] for details concerning this (long!) construction. The result of this construction is the following lemma. We remark that, for the construction of linkings of higher order, the functional space needs to have a sufficiently large dimension. Then, the Lyapunov-Schmidt reduction gives a critical point for P^ if a critical point for Q^ is found. Then we may take advantage of the Morse-type information (2.17) to find an upper bound for the eigenvalues e^ strictly lower than 1. Now, it is possible, thanks to our compactness results, to pass to the limit first as the pulsation u goes to 0: then, J[u} -> +00 and we obtain a sequence (^)j>i of critical points for the functional ^p, with eigenvalues ^. Secondly, we may pass to the limit as the parameters v and p go respectively to 0 and oo, which solves the initial problem.